Convert Object texture coords to Polar
You can convert Object coordinates into Polar coordinates using the 'arctangent' trigonometric function by using :

arctangent(y/x)


This will produce a result in Radians (where there are pi radians in 360 degrees) although without additional nodes it will only produce results for half the rotation (and will produce negative values).

The following nodes should produce a result that varies from 0.0 to 1.0 :



To explain this a bit, the Object coordinates are split into the separate X, Y, Z components and the Divide node used to generate Y / X. This is passed to the Arctangent node which produces the angle (in radians) between -pi/2 and pi/2 for half a rotation (for the second half it repeats -pi/2 to pi/2). This is divided by 2*pi to convert it to the range -0.25 to 0.25 and 0.25 added to shift the range to 0.0 to 0.5. The Less Than node is then used to detect the second half of the rotation and add 0.5, resulting in the full rotation mapping from 0.0 to 1.0 producing the result shown in the image above.

Blend file included 



The other half of the polar coordinates is the distance from the origin. This is simple to achieve as sqrt(x^2 + y^2) as follows :



Combining these into a nodegroup to allow it to be easily reused results in the following :



Using the nodegroup to build a procedural eye texture can produce the following :





Blend file included